Laplacians on periodic graphs with guides

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Laplacians on periodic graphs with guides Evgeny Korotyaev a , Natalia Saburova b,∗ a b

Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia Northern (Arctic) Federal University, Severnaya Dvina Emb. 17, Arkhangelsk, 163002, Russia

a r t i c l e

i n f o

Article history: Received 6 February 2017 Available online xxxx Submitted by P. Exner Keywords: Discrete Laplace operator Periodic graph Guided waves

a b s t r a c t We consider Laplace operators on periodic discrete graphs perturbed by guides, i.e., graphs which are periodic in some directions and finite in other ones. The spectrum of the Laplacian on the unperturbed graph is a union of a finite number of nondegenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed Laplacian consists of the unperturbed one plus the additional socalled guided spectrum which is a union of a finite number of bands. We estimate the position of the guided bands and their length in terms of geometric parameters of the graph. We also determine the asymptotics of the guided bands for guides with large multiplicity of edges. Moreover, we show that the possible number of guided bands, their length and position can be rather arbitrary for some specific periodic graphs with guides. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Laplacians on periodic discrete graphs have attracted a lot of attention due to their applications to the study of electronic properties of real crystalline structures, see, e.g., [11,32] and the survey [6]. However, the arrangement of atoms or molecules in most crystalline materials is not perfect. The regular patterns are interrupted by crystalline defects. These defects are the most important features of the engineering material and are manipulated to control its behavior. We consider Laplace operators on periodic discrete graphs perturbed by guides (i.e., graphs which are periodic in some directions and finite in others). For example, a guide is a periodic graph embedded into a strip in the case of planar graphs. It is well known that the spectrum of discrete Laplacians on periodic graphs has a band structure with a finite number of flat bands (eigenvalues of infinite multiplicity) [13,15, 23,35]. The spectrum of the Laplacian on the perturbed graph consists of the spectrum of the Laplacian on the unperturbed periodic graph plus the so-called guided spectrum. The additional guided spectrum is * Corresponding author. E-mail addresses: [email protected], [email protected] (E. Korotyaev), [email protected], [email protected] (N. Saburova). http://dx.doi.org/10.1016/j.jmaa.2017.06.039 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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a union of a finite number of bands and the corresponding wave-functions are mainly located along the guides. In our paper we study guided spectra of Laplacians. We describe our main goals: • to estimate the position of guided bands and their lengths in terms of geometric parameters of graphs; • to determine asymptotics of the guided spectrum for guides with large multiplicity of edges; • to show that a possible number of guided bands (including flat bands), their length and positions can be rather arbitrary for some specific periodic graphs with guides. 1.1. Discrete Laplacians on periodic graphs Let Γ = (V, E) be a connected infinite graph, possibly having loops and multiple edges, where V is the set of its vertices and E is the set of its unoriented edges. From the set E we construct the set A of oriented edges by considering each edge in E to have two orientations. An edge starting at a vertex u and ending at a vertex v from V will be denoted as the ordered pair (u, v) ∈ A. Vertices u, v ∈ V will be called adjacent and denoted by u ∼ v, if (u, v) ∈ A. We define the degree κv of the vertex v ∈ V as the number of all edges from A starting at v. We consider graphs with uniformly bounded degrees. Let 2 (V ) be the Hilbert space of all functions f : V → C equipped with the norm f 22 (V ) =



|f (v)|2 < ∞.

v∈V

We define the discrete Laplacian (i.e., the combinatorial Laplace operator) Δ on 2 (V ) by   Δf (v) =





 f (v) − f (u) ,

f = (f (v))v∈V ∈ 2 (V ),

(1.1)

e=(v,u)∈A

where the sum is taken over all oriented edges starting at the vertex v ∈ V . It is well known, see, e.g., [31], that Δ is self-adjoint and its spectrum satisfies: the point 0 belongs to the spectrum σ(Δ) containing in [0, 2κ+ ], i.e., 0 ∈ σ(Δ) ⊂ [0, 2κ+ ],

κ+ = sup κv < ∞.

where

v∈V

(1.2)



We consider a Zd -periodic graph Γ0 = (V0 , E0 ), i.e., a graph satisfying the following conditions: 

1) Γ0 is equipped with an action of the free abelian group Zd ;  2) the quotient graph Γ∗ = (V∗ , E∗ ) = Γ0 /Zd is finite. We assume that the graphs are embedded into Euclidean space, since in many applications such a natural embedding exists. For example, in the tight-binding approximation real crystalline structures are modeled as discrete graphs embedded into Rd (d = 2, 3) and consisting of vertices (points representing positions of atoms) and edges (representing chemical bonding of atoms), by ignoring the physical characters of atoms and bonds that may be different from one another. But all results of the paper stay valid in the case of abstract periodic graphs (without the assumption of graph embedding into Euclidean space).   For a periodic graph Γ0 embedded into the space Rd , the quotient graph Γ0 /Zd is a graph on the    torus Rd /Zd . Due to the definition, the graph Γ0 is invariant under translations through d-dimensional  vectors a1 , . . . , ad which generate the group Zd : Γ0 + as = Γ0 ,

∀ s ∈ Nd .

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Here and below for each integer m the set Nm is given by Nm = {1, . . . , m }.

(1.3) 

We will call the vectors a1 , . . . , ad the periods of the graph Γ0 . In the space Rd we consider a coordinate system with the origin at some point O and with the basis a1 , . . . , ad. Below the coordinates of all graph vertices will be expressed in this coordinate system. Let A be a self-adjoint operator, we denote by σ(A), σess (A), σac (A), σp (A), and σf b (A) its spectrum, essential spectrum, absolutely continuous spectrum, point spectrum (eigenvalues of finite multiplicity), and the flat band spectrum (eigenvalues of infinite multiplicity), respectively. We consider the Laplacian defined by (1.1) on the periodic graph Γ0 as an unperturbed operator and denote it by Δ0 . It is well known that the spectrum σ(Δ0 ) of the Laplacian on periodic graphs is a union of ν spectral bands σn (Δ0 ): σ(Δ0 ) =

ν 

σn (Δ0 ) = σac (Δ0 ) ∪ σf b (Δ0 ),

(1.4)

n=1

where ν = #V∗ is the number of vertices of the quotient graph Γ∗ , the absolutely continuous spectrum σac (Δ0 ) consists of non-degenerate bands σn (Δ0 ). Note that each flat band is a degenerate band. The spectrum σ(Δ0 ) is a subset of the interval [0, ]: σ(Δ0 ) ⊂ [0, ],

inf σ(Δ0 ) = 0,

sup σ(Δ0 ) = .

(1.5)

1.2. Results overview There are results about spectral properties of the Schrödinger operator H0 = Δ0 + W with a periodic potential W . The decomposition of the operator H0 into a constant fiber direct integral was obtained in [13,15,35] without an exact form of fiber operators and in [23,27] with an exact form of fiber operators. In particular, this yields the band-gap structure of the spectrum of the operator H0 . In [9] the authors described different properties of Schrödinger operators with periodic potentials on the lattice Z2 , the simplest Z2 -periodic graph. In [24,30] the positions of the spectral bands of the Laplacians were estimated in terms of eigenvalues of the operator on finite graphs (the so-called eigenvalue bracketing). The estimate of the total length of all bands σn (H0 ) given by ν 

|σn (H0 )|  2β,

(1.6)

n=1

was obtained in [23]; where β = #E∗ + 1 − ν is the so-called Betti number, #E∗ is the number of edges of the quotient graph Γ∗ . Moreover, a global variation of the Lebesgue measure of the spectrum and a global variation of the gap-length in terms of potentials and geometric parameters of the graph were determined. Note that the estimate (1.6) also holds true for magnetic Schrödinger operators with periodic magnetic and electric potentials (see [27]). Estimates of the Lebesgue measure of the spectrum of H0 in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph were described in [24]. Estimates of effective masses, associated with the ends of each spectral band of Laplacians, in terms of geometric parameters of the graphs were obtained in [25]. Moreover, in the case of the bottom of the spectrum two-sided estimates on the effective mass in terms of geometric parameters of the graphs were determined. The proof of all these results in [23–25,27] is based on Floquet theory and the exact form of fiber Schrödinger operators from [23,27]. The spectra of the discrete Schrödinger operators on graphene nano-tubes and nano-ribbons in external fields were discussed in [20,21]. The spectrum of discrete magnetic

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Laplacians on some planar graphs (the hexagonal lattice, the kagome lattice and so on) was described in [12] and see the references therein. Discrete Laplacians for some class of periodic graphs with compact perturbations including the square, triangular, diamond, kagome lattices were discussed in [2]. Laplacians on periodic graphs with non-compact perturbations and the stability of their essential spectrum were considered in [38]. The spectrum of Laplacians on the lattice Zd with pendant edges was studied in [40]. In the paper [26], the authors considered Schrödinger operators with periodic potentials on periodic discrete graphs perturbed by so-called guided potentials, which are periodic in some directions and finitely supported in others. They described some properties of the additional guided spectrum. We remark that the case of guided potentials is simpler than the case of periodic graphs with guides and helps us to understand better the properties of the guided spectrum in the case of periodic graphs with guides. It is important that in the case of guided potentials all operators act in the same space. But in the case of periodic graphs with guides this is not true. Note that line defects on the lattice were considered in [7,28,29,33]. Scattering theory for self-adjoint Schrödinger operators with decreasing potentials was investigated in [5,17] (for the lattice) and in [34] (for periodic graphs). Inverse scattering theory with finitely supported potentials was considered in [17] for the case of the lattice Zd and in [1] for the case of the hexagonal lattice. The absence of eigenvalues embedded in the essential spectrum of the operators was discussed in [18,41]. Trace formulae and global eigenvalues estimates for Schrödinger operators with complex decaying potentials on the lattice were obtained in [22]. The Cwikel–Lieb–Rosenblum type bound for the discrete Schrödinger operator on Zd was computed in [19,37]. Finally, we note that different properties of Schrödinger operators on graphs were considered in [10,39]. 2. Main results 2.1. The unperturbed case: periodic graphs 

For a given number d ∈ Nd−1 we define the infinite fundamental graph C0 of the Zd -periodic graph Γ0  by C0 = (V0c , E0c ) = Γ0 /Zd ,

E0c = E0 /Zd ,

V0c = V0 /Zd , 

where Zd is considered as the subgroup Zd ×{0} ×. . .×{0} of Zd . The graph C0 has the vertex set V0c and the   set E0c of unoriented edges. Remark that the graph C0 is a graph on the cylinder Rd /Zd and is Zd−d -periodic. We also call the fundamental graph C0 a discrete cylinder or just a cylinder. We identify the vertices of the  cylinder C0 with the vertices of the periodic graph Γ0 from the fundamental strip S = [0, 1)d × Rd−d . We will call this infinite vertex set a fundamental vertex set of Γ0 and denote it by the same symbol V0c : V0c = V0 ∩ S,



S = [0, 1)d × Rd−d .

(2.1)

Edges of the periodic graph Γ0 connecting the vertices from the fundamental vertex set V0c with the vertices from V0 \ V0c will be called bridges. Bridges always exist and provide the connectivity of the periodic graph. We denote the set of all bridges of the graph Γ0 by B. 2.2. The perturbed case: periodic graphs with guides We define the union of two graphs G0 = (V0 , E0 ) and G1 = (V1 , E1 ) as the graph G given by G = G0 ∪ G1 = (V , E ),

V = V0 ∪ V1 ,

E = E0 ∪ E1 .

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Fig. 1. a) The square lattice L2 ; the vertices from the set V0c are big black points; the fundamental strip S is shaded; b) the unperturbed cylinder C0 = L2 /Z (the edges of the strip are identified); c) a perturbation Γ1 with two connected components; d) the perturbed square lattice Γ = L2 ∪ Γg1 ; e) the perturbed cylinder C = Γ/Z (the edges of the strip are identified). 

Recall that the graphs under consideration are embedded into the space Rd and hence vertices of the graphs G0 and G1 may be the same in the embedding. Now we define a periodic graph with guides. Let Γ1 = (V1 , E1 ) be a finite graph, possibly not connected, such that all vertices of Γ1 are contained in the fundamental strip S and the graph Γ0 ∪ Γ1 is connected. A Zd -periodic graph 

Γg1 =

(Γ1 + m)

(2.2)

m∈Zd

will be called a guide with the fundamental graph Γ1 . We define a perturbed graph Γ as a union of the unperturbed periodic graph Γ0 and the perturbation Γg1 : Γ = Γ0 ∪ Γg1 .

(2.3)

We will call the graph Γ a periodic graph with a guide Γg1 or a perturbed graph. Due to the definition (2.3) of the perturbed graph Γ, the perturbed cylinder C = Γ/Zd = (V c , E c ) for Γ is a union of the unperturbed cylinder C0 = (V0c , E0c ) and the finite graph Γ1 = (V1 , E1 ): C = C0 ∪ Γ1 ,

(2.4)

i.e., V c = V0c ∪ V1 ,

E c = E0c ∪ E1 ,

Ac = Ac0 ∪ A1 ,

(2.5)

where Ac , Ac0 and A1 are the sets of all doubled oriented edges of C, C0 and Γ1 , respectively. Example. For the square lattice L2 with the periods a1 , a2 , see Fig. 1.a, the unperturbed cylinder C0 = L2 /Z = (V0c , E0c ) is shown in Fig. 1.b. The vertices from the set V0c are big black points in Fig. 1.a. A perturbation Γ1 , the perturbed square lattice Γ = L2 ∪ Γg1 and the perturbed cylinder C = Γ/Z are shown in Fig. 1.c,d,e. 2.3. Floquet decomposition and the spectrum of perturbed Laplacians We describe the basic spectral properties of Laplacians on periodic graphs with guides. Proposition 2.1. i) The Laplacian Δ on a perturbed graph Γ has the following decomposition into a constant fiber direct integral for some unitary operator U : 2 (V ) → H : ⊕ H =

dϑ  (V ) , (2π)d 2

Td

c

U ΔU

−1

⊕ =

Δ(ϑ) Td

dϑ , (2π)d

(2.6)

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where Td = Rd /(2πZ)d and the fiber Laplacian Δ(ϑ) on the fiber space 2 (V c ) is given by 

 Δ(ϑ)f (v) =





 f (v) − eiτ (e), ϑ f (u) ,

v ∈ V c,

f ∈ 2 (V c ).

(2.7)

e=(v, u)∈Ac

Here τ (e) ∈ Zd is the index of the edge e ∈ Ac defined by (3.3), (3.4), V c and Ac are the vertex set and the set of oriented edges of the cylinder C, respectively; · , · is the inner product in Rd . ii) For each ϑ ∈ Td the spectrum of the fiber operator Δ(ϑ) has the form         σ Δ(ϑ) = σac Δ(ϑ) ∪ σf b Δ(ϑ) ∪ σp Δ(ϑ) ,         σac Δ(ϑ) = σac Δ0 (ϑ) , σf b Δ(ϑ) = σf b Δ0 (ϑ) ,

(2.8) (2.9)

  where Δ0 (ϑ) is the fiber operator for the unperturbed Laplacian Δ0 on the periodic graph Γ0 , σp Δ(ϑ) is the set of all eigenvalues of Δ(ϑ) of finite multiplicity given by λNϑ (ϑ)  . . .  λ2 (ϑ)  λ1 (ϑ),

Nϑ  p := rank Δ1 = ν1 − cΓ1 ,

ν1 = #V1 ,

(2.10)

Δ1 is the Laplacian on the finite graph Γ1 = (V1 , E1 ) and cΓ1 is the number of connected components of Γ1 , #A denotes the number of all elements of the set A. Remark. The fiber Laplacian Δ(ϑ), ϑ ∈ Td , can be considered as a magnetic Laplacian on the cylinder C (see [14,27]). Proposition 2.1 and standard arguments (see Theorem XIII.85 in [36]) describe the spectrum of the Laplacian Δ on a perturbed graph Γ. Since Δ(ϑ) is self-adjoint and real analytic on the torus Td = Rd /(2πZ)d , each λj (·) is a real and piecewise analytic function on Td and creates the guided band sj (Δ) given by + d sj (Δ) = [λ− j , λj ] = λj (T ),

j = 1, . . . , N,

N = max Nϑ  p. ϑ∈Td

(2.11)

Thus, the spectrum of the Laplacian Δ on the perturbed graph Γ has the form σ(Δ) =



  σ Δ(ϑ) = σ(Δ0 ) ∪ s(Δ),

ϑ∈Td

where σ(Δ0 ) is defined by (1.4) and s(Δ) =

 ϑ∈Td

N    σp Δ(ϑ) = sj (Δ) = sac (Δ) ∪ sf b (Δ),

(2.12)

j=1

sac (Δ) and sf b (Δ) are the absolutely continuous part and the flat band part of the guided spectrum s(Δ), respectively. An open interval between two neighboring non-degenerate bands is called a spectral gap. The guided spectrum s(Δ) may partly lie above the spectrum of the unperturbed operator Δ0 , on the spectrum of Δ0 and in the gaps of Δ0 . We formulate simple sufficient conditions for the existence of guided flat bands of the perturbed Laplacian Δ. Proposition 2.2. Let Γ0 be a periodic graph. Assume that ζ is an eigenvalue of the Laplacian Δ1 on a finite graph Γ1 with an eigenfunction f ∈ 2 (V1 ) equal to zero on V0c ∩ V1 , i.e.,

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Fig. 2. a) The perturbed square lattice Γ = L2 ∪ Γg1 ; b) The finite graph Γ1 ; c) the perturbed cylinder C = Γ/Z; d) the spectra of the fiber Laplacians Δ(ϑ) as ϑ = 0, π and the Laplacian Δ.

∀v ∈ V01 = V0c ∩ V1 .

f (v) = 0,

(2.13)

Then {ζ} is a guided flat band of the Laplacian Δ on the perturbed graph Γ = Γ0 ∪ Γg1 . Example. We consider the perturbed square lattice Γ = L2 ∪ Γg1 shown in Fig. 2.a. For each ϑ ∈ T = (−π, π] the spectrum of the fiber Laplacian Δ(ϑ) has the form       σ Δ(ϑ) = σac Δ(ϑ) ∪ σp Δ(ϑ) ,

    σac Δ(ϑ) = σ Δ0 (ϑ) = [2 − 2 cos ϑ, 6 − 2 cos ϑ],

  σp Δ(ϑ) consists of two eigenvalues λ1 (ϑ) and λ2 (ϑ) = 3, see Fig. 2.d. The spectrum of the Laplacian Δ on Γ has the form σ(Δ) = σ(Δ0 ) ∪ s(Δ),

σ(Δ0 ) = [0, 8],

where the guided spectrum s(Δ) is given by (see Fig. 2.d and details in Proposition 5.3) s(Δ) = sac (Δ) ∪ sf b (Δ),

sac (Δ) = s1 (Δ) = λ1 (T) ≈ [10,6; 13,9], sf b (Δ) = s2 (Δ) = λ2 (T) = {3}.

The Laplacian Δ1 on the finite graph Γ1 has the eigenvalue λ = 3 with an eigenfunction f such that f (0) = 0. Then, due to Proposition 2.2, {3} is a guided flat band of Δ on the perturbed square lattice Γ = L2 ∪ Γg1 . Here we remark that we do not know an example of a Schrödinger operator with a guided potential on a periodic graph having a guided flat band. 2.4. Estimates of guided bands We consider the guided bands from (2.11) (or their parts) above the spectrum of the unperturbed Laplacian Δ0 : soj (Δ) = sj (Δ) ∩ [, +∞) = ∅,

j = 1, . . . , Ng ,

Ng  N  p,

(2.14)

recall that  = sup σ(Δ0 ). The Laplacian Δ1 on the finite graph Γ1 = (V1 , E1 ) has the eigenvalue 0 of multiplicity cΓ1 and p positive eigenvalues ζj labeled by 0 < ζp  . . .  ζ2  ζ1 ,

p = ν1 − cΓ1 ,

ν1 = #V1 ,

counting multiplicity, where cΓ1 is the number of connected components of the graph Γ1 .

(2.15)

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Proposition 2.1 and the standard perturbation theory give the estimates of the position of the bands soj (Δ) and their number Ng by (for more details see Corollary 4.1) soj (Δ) ⊂ [ζj , ζj + ],

Ng  #{j ∈ Np : ζj > },

(2.16)

where #A is the number of elements of the set A. In particular, this yields that if the eigenvalues of Δ1 satisfy ζp >  and ζj − ζj+1 >  for all j ∈ Np−1 , then the guided spectrum of the Laplacian Δ consists of exactly p guided bands separated by gaps. In order to formulate our main result we define the set Bc = B/Zd of all bridges of the cylinder C = (V c , E c ) and the modified cylinder C m = (V c , E c \B c ), which is obtained from C by deleting all its bridges. We consider the Laplacian Δm defined by (1.1) on the modified cylinder C m . This Laplacian Δm has at most p = rank Δ1 eigenvalues μ 1  μ 2  . . . . Define μj by μj = max{ μj , sup σess (Δm )},

j = 1, 2, . . . , p.

(2.17)

We estimate the position of the guided bands soj (Δ) defined by (2.14) in terms of the eigenvalues of the operator Δm and the number of bridges on the cylinder C. Theorem 2.3. i) Let Δ be the Laplacian on a perturbed graph Γ and let μj be defined by (2.17). Then each guided band soj (Δ), j = 1, . . . , Ng , defined by (2.14) satisfies soj (Δ) ⊂ [μj , μj + 2β+ ],

β+ = maxc βv , v∈V

(2.18)

where βv is the number of bridges on C starting at the vertex v ∈ V c . ii) Moreover, for any ε > 0 there exists a perturbed graph Γ such that each non-degenerate guided band length |soj (Δ)| > 2β+ − ε, j = 1, . . . , Ng . Remark. For most of graphs the number β+ = 1, then the guided band length |soj (Δ)|  2 for all j = 1, . . . , Ng , but for specific graphs β+ may be any given integer number. Let Γt = (V1 , Et ) be a finite graph obtained from the graph Γ1 = (V1 , E1 ) considering each edge of Γ1 to have the multiplicity t ∈ N. We consider the Laplacian Δt acting on a perturbed graph Γ = Γ0 ∪ Γgt and discuss the guided spectrum of Δt for large t. Theorem 2.4. Let Δt be the Laplacian on the perturbed graph Γ = Γ0 ∪ Γgt , where Γ0 is any periodic graph and t ∈ N is large enough. Then the guided spectrum of the Laplacian Δt consists of exactly p guided bands, where p is defined in (2.15), and the following statements hold true: i) Let ζj for some j ∈ Np be a simple positive eigenvalue of the Laplacian Δ1 on the graph Γ1 with a normalized real-valued eigenfunction fj ∈ 2 (V1 ). • If fj = 0 on V01 = V0c ∩ V1 , then {tζj } is a guided flat band of the Laplacian Δt . + • If fj = 0 on V01 , then the guided band sj (Δt ) = [λ− j (t), λj (t)] satisfies ± λ± j (t) = tζj + Wj + O(1/t),

|sj (Δt )| = Wj• + O(1/t),

(2.19)

as t → ∞, where Wj− = min Wj (ϑ), ϑ∈Td

Wj•

=

Wj+

−

Wj− ,

Wj+ = max Wj (ϑ), ϑ∈Td

Wj•

 2β01 ,

(2.20)

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for some function Wj defined by the formula (4.13). Here β01 is the number of all oriented bridges connecting the vertices from V01 on the cylinder C. ii) In particular, if the set V01 consists of one vertex v, then fj2 (v)β01  Wj•  2fj2 (v)β01 .

(2.21)

Moreover, Wj• = 0 iff β01 = 0. iii) Let all positive eigenvalues ζj , j ∈ Np , of Δ1 be distinct. Then all guided bands of the Laplacian Δt are separated by gaps and the Lebesgue measure |s(Δt )| of the guided spectrum of Δt satisfies |s(Δt )| =

p 

Wj• + O(1/t).

(2.22)

j=1

iv) In particular, if there is no bridge connecting the vertices from the set V01 on the cylinder C, then = 0 for each j ∈ Np and the second identity in (2.19) and the formula (2.22) take the form |sj (Δt )| = O(1/t), j ∈ Np , and |s(Δt )| = O(1/t), respectively.

Wj•

Now we describe geometric properties of the guided spectrum for periodic graphs with specific guides. Corollary 2.5. Let Γ0 be a periodic graph with an unperturbed cylinder C0 = (V0c , E0c ). Then the following statements hold true. i) For any constant C > 0 there exists a finite graph Γt , t ∈ N, such that the Lebesgue measure of the guided spectrum s(Δ) of the perturbed Laplacian Δ on Γ = Γ0 ∪ Γgt satisfies |s(Δ)| > C and all guided bands are non-degenerate. ii) Let, in addition, there exist a vertex v ∈ V0c such that there is no bridge on C0 starting at v. Then for any small ε > 0 there exists a finite graph Γt such that the Lebesgue measure of the guided spectrum s(Δ) of the perturbed Laplacian Δ on Γ = Γ0 ∪ Γgt satisfies |s(Δ)| < ε. iii) For any constant λ0 > 0 there exists a finite graph Γt such that the guided spectrum s(Δ) of the perturbed Laplacian Δ on Γ = Γ0 ∪ Γgt satisfies s(Δ) ∩ (λ0 , +∞) = ∅. iv) There exists a finite graph Γ1 such that the guided spectrum s(Δ) of the perturbed Laplacian Δ on Γ = Γ0 ∪ Γg1 has a degenerate guided band. Thus, roughly speaking, the guided spectrum can be any set above the unperturbed spectrum. Its Lebesgue measure can be arbitrarily large or arbitrarily small. We present the plan of our paper. In Section 3 we introduce the notion of edge indices and prove Proposition 2.1 about the decomposition of the Laplacian on periodic graphs with guides into a constant fiber direct integral. In Section 4 we prove Theorems 2.3, 2.4 and Corollary 2.5. Section 5 is devoted to properties of the guided spectrum for the square lattice with specific guides. 3. Direct integral for Laplacians on periodic graphs with guides 3.1. Edge indices In order to give a decomposition of Laplacians on periodic graphs with guides into a constant fiber direct integral with a precise representation of fiber operators we need to define an edge index. Recall that an edge index was introduced in [23] and it was important to study the spectrum of Laplacians and Schrödinger operators on periodic graphs, since fiber operators are expressed in terms of edge indices (see (2.7)).

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For any v ∈ V the following unique representation holds true: v0 ∈ V c ,

v = v0 + [v],

[v] ∈ Zd ,

(3.1)

where V c is the fundamental vertex set of the graph Γ = (V, E) defined by V c = V ∩ S,



S = [0, 1)d × Rd−d .

(3.2)

Recall that the coordinates of graph vertices are taken with respect to the basis a1 , . . . , ad and Zd is  considered as the subgroup Zd × {0} × . . . × {0} of Zd . In other words, each vertex v can be obtained from a vertex v0 ∈ V c by the shift by a vector [v] ∈ Zd . For any oriented edge e = (u, v) ∈ A we define the edge “index” τ (e) as the integer vector given by τ (e) = [v] − [u] ∈ Zd ,

(3.3)

where, due to (3.1), we have u = u0 + [u],

v = v0 + [v],

u0 , v0 ∈ V c ,

[u], [v] ∈ Zd .

We note that edges connecting vertices from the fundamental vertex set V c have zero indices. We define a surjection fA : A → Ac = A/Zd , which map each edge to its equivalence class. If e is an oriented edge of the graph Γ, then there is an oriented edge e∗ = fA (e) on the cylinder C = Γ/Zd . For the edge e∗ ∈ Ac we define the edge index τ (e∗ ) by τ (e∗ ) = τ (e).

(3.4)

In other words, edge indices of the cylinder C are induced by edge indices of the graph Γ. Edges with nonzero indices are called bridges. Edge indices, generally speaking, depend on the choice of the coordinate origin O and the periods a1 , . . . , ad of the graph Γ or, in other words, on the choice of the fundamental strip S. But in a fixed coordinate system indices of the cylinder edges are uniquely determined by (3.4), since τ (e + m) = τ (e),

∀ (e, m) ∈ A × Zd .

We note that, due to the definition of periodic graphs with guides, τ (e) = 0,

∀ e ∈ A1 .

(3.5)

3.2. Direct integrals We prove Proposition 2.1 about the decomposition of Laplacians on periodic graphs with guides into a constant fiber direct integral. Proof of Proposition 2.1.i). Repeating the arguments from the proof of Theorem 1.1 in [23] we obtain (2.6), (2.7), where the unitary operator U : 2 (V ) → H has the form (U f )(ϑ, v) =



e−im,ϑ f (v + m),

(ϑ, v) ∈ Td × V c ,

f ∈ 2 (V ).

(3.6)

m∈Zd

The Hilbert space H defined in (2.6) is equipped with the norm g2H = the function g(ϑ, ·) ∈ 2 (V c ) for almost all ϑ ∈ Td . 2

 Td

dϑ g(ϑ, ·)22 (V c ) (2π) d , where

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In order to prove the next item of Proposition 2.1 we need the following lemma. Lemma 3.1. Let P and P1 be the orthogonal projections of 2 (V c ) onto the subspaces 2 (V0c ) and 2 (V1 ), respectively. Then each fiber Laplacian Δ(ϑ), ϑ ∈ Td , defined by (2.7) has the following decomposition: Δ(ϑ) = P Δ0 (ϑ)P + P1 Δ1 P1 ,

(3.7)

where Δ0 (ϑ) is the fiber operator for the unperturbed Laplacian Δ0 on the periodic graph Γ0 , Δ1 is the Laplacian on the finite graph Γ1 = (V1 , E1 ). Proof. The Laplacian Δ0 on the unperturbed periodic graph Γ0 = (V0 , E0 ) has a decomposition into a constant fiber direct integral for some unitary operator U0 : 2 (V0 ) → H0 : ⊕ H0 =

2 (V0c )

dϑ , (2π)d

U0 Δ0 U0−1 =

Td

⊕ Δ0 (ϑ)

dϑ , (2π)d

(3.8)

Td

where the fiber Laplacian Δ0 (ϑ) acts on the fiber space 2 (V0c ) and is given by 

  Δ0 (ϑ)f0 (v) =

e=(v,



 f0 (v) − eiτ (e), ϑ f0 (u) ,

v ∈ V0c ,

f0 ∈ 2 (V0c ).

(3.9)

u)∈Ac0

For each f ∈ 2 (V c ) we have

(P Δ0 (ϑ)P + P1 Δ1 P1 )f, f V c = Δ0 (ϑ)P f, P f V0c + Δ1 P1 f, P1 f V1         = Δ0 (ϑ)P f (v) f (v) + Δ1 P1 f (v) f (v) = Δ0 (ϑ)P f (v) f (v)



v∈V0c

+

v∈V0c \V01

v∈V1



     Δ1 P1 f (v) f (v) + (Δ0 (ϑ)P + Δ1 P1 )f (v) f (v),

v∈V1 \V01

v∈V01

where ·, · V denotes the inner product in 2 (V ). Substituting the definitions (1.1) and (3.9) of the Laplacian Δ1 and the fiber Laplacian Δ0 (ϑ) into this formula and using the identities (2.5), (3.5) and (2.7), we obtain

(P Δ0 (ϑ)P + P1 Δ1 P1 )f, f V c =     = f (v) − eiτ (e), ϑ f (u) f (v) + v∈V0c \V01 e=(v, u)∈Ac0

+

 

v∈V01

=



 e=(v, u)∈Ac0



v∈V c e=(v, u)∈Ac

which implies (3.7).









 f (v) − f (u) f (v)

v∈V1 \V01 (v,u)∈A1

f (v) − e

iτ (e), ϑ



f (u) +





f (v) − f (u)



f (v)

(v,u)∈A1

     f (v) − eiτ (e), ϑ f (u) f (v) = Δ(ϑ)f (v) f (v) = Δ(ϑ)f, f V c , v∈V c

2 

Proof of Proposition 2.1.ii). For each ϑ ∈ Td the unperturbed fiber Laplacian Δ0 (ϑ) is Zd−d -periodic. Then, using standard arguments (see Theorem XIII.85 in [36]), we obtain that the spectrum of Δ0 (ϑ) has the form       σ Δ0 (ϑ) = σac Δ0 (ϑ) ∪ σf b Δ0 (ϑ) .

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  Since the graph Γ1 is finite, for each λ ∈ σf b Δ0 (ϑ) there exists a corresponding eigenfunction f ∈ 2 (V0c ) with a finite support (see, e.g., Theorem 4.5.2 in [4]) not intersecting with V1 . Due to (3.7), (f, 0) ∈ 2 (V c )   is an eigenfunction of Δ(ϑ) with the same finite support and the same eigenvalue λ. Thus, λ ∈ σf b Δ(ϑ) and vice versa. Since the operator Δ1 has finite rank p, where p is defined in (2.10), for each ϑ ∈ Td the       spectrum σ Δ(ϑ) of the fiber Laplacian Δ(ϑ) is given by (2.8), where σac Δ(ϑ) , σf b Δ(ϑ) satisfy (2.9)   and σp Δ(ϑ) consists of Nϑ  p eigenvalues (2.10). 2 4. Proof of the main results In this section we prove Theorem 2.3 about the position of guided bands and Theorem 2.4 about the asymptotics of the guided bands for guides with large multiplicity of their edges. We prove Corollary 2.5 about geometric properties of the guided spectrum for periodic graphs with specific guides. 4.1. Estimates for the guided spectrum Denote by m± (ϑ) the upper and lower endpoints of the spectrum of the unperturbed fiber Laplacian Δ0 (ϑ):   m− (ϑ) = inf σ Δ0 (ϑ) ,

  m+ (ϑ) = sup σ Δ0 (ϑ) .

(4.1)

max m+ (ϑ) = .

(4.2)

Then (1.5) yields min m− (ϑ) = 0,

ϑ∈Td

ϑ∈Td

We need a simple estimate for eigenvalues of bounded self-adjoint operators [36]: Let A, B be bounded

 j (A), sup σess (A) , j = 1, 2, . . . , where self-adjoint operators in a Hilbert space H and let λj (A) = max λ 2 (A)  . . . are the eigenvalues of A. Then 1 (A)  λ λ λj (A) + inf σ(B)  λj (A + B)  λj (A) + sup σ(B),

j = 1, 2, 3, . . . .

(4.3)

The following simple corollary about the position of the guided bands soj (Δ) defined by (2.14) is a direct consequence of Proposition 2.1. Corollary 4.1. Let Δ be the Laplacian on a perturbed graph Γ and let  = sup σ(Δ0 ). Then each guided band soj (Δ), j = 1, . . . , Ng , and their number Ng satisfy soj (Δ) ⊂ [ζj , ζj + ],

(4.4)

Ng  #{j ∈ Np : ζj > },

(4.5)

where ζ1  . . .  ζp are the positive eigenvalues of the Laplacian Δ1 and p = rank Δ1 . Proof. Each fiber Laplacian Δ(ϑ), ϑ ∈ Td , is given by (3.7). Then, due to (4.1)–(4.3), each eigenvalue λj (ϑ) of Δ(ϑ) above its essential spectrum satisfies ζj  ζj + m− (ϑ)  λj (ϑ)  ζj + m+ (ϑ)  ζj + ,

(4.6)

which yields (4.4). Let ζj >  for some j = 1, . . . , p. Then, due to (4.6), λj (ϑ) >  for all ϑ ∈ Td . Thus, λj creates the guided band soj (Δ) = λj (Td ). This yields (4.5). 2

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Remark. It is well known, see, e.g., [8], that the positive eigenvalues ζ1  . . .  ζp of the Laplacian Δ1 on a finite graph Γ1 = (V1 , E1 ) satisfy ν1 max κ 1  ζ1  max (κu1 + κv1 ), u,v∈V1 ν1 − 1 v∈V1 v u∼v

min

u,v∈V1 u∼v

(κu1

+

κv1 )

− (ν1 − 2)  ζp 

ν1 min κ 1 , ν1 − 1 v∈V1 v

where ν1 = #V1 , κv1 is the degree of the vertex v ∈ V1 on Γ1 . From these estimates and (4.4) it follows that the first guided band so1 (Δ) moves arbitrarily far to the right with increase in the degree of at least one vertex of the graph Γ1 . All guided bands move arbitrarily far to the right with increase in the degree of all vertices of Γ1 . Proof of Theorem 2.3. i) We rewrite the fiber Laplacian Δ(ϑ), ϑ ∈ Td , defined by (2.7) in the form: 



Δβ (ϑ)f (v) =

Δ(ϑ) = Δm + Δβ (ϑ),    f (v) − eiτ (e), ϑ f (u) , e=(v, u)∈A τ (e) =0

(4.7) v ∈ V c,

(4.8)

c

where τ (e) ∈ Zd is the index of the edge e ∈ Ac defined by (3.3), (3.4). Each operator Δβ (ϑ), ϑ ∈ Td , is the magnetic Laplacian on the graph Cβ = (V c , B c ) and the degree of each vertex v ∈ V c on Cβ is   equal to the number βv of all bridges starting at v. Then the spectrum σ Δβ (ϑ) ⊂ [0, 2β+ ] for each ϑ ∈ Td (see, e.g., [14]) and, due to (4.3), each eigenvalue λj (ϑ) of Δ(ϑ) above its essential spectrum satisfies μj  λj (ϑ)  μj + 2β+ , which yields (2.18). ii) The existence of such graphs is proved in Proposition 5.2. 2 4.2. Proof of Theorem 2.4 Let Γt = (V1 , Et ) be a finite graph obtained from the graph Γ1 = (V1 , E1 ) considering each edge of Γ1 to have the multiplicity t ∈ N. Then the Laplacian on the graph Γt has the form tΔ1 . If t is large enough, then all positive eigenvalues tζp  . . .  tζ1 of the Laplacian tΔ1 on the graph Γt satisfy tζj >  for all j ∈ Np , where  is defined in (1.5). Then, due to the inequalities Ng  p and (4.5), the guided spectrum of the Laplacian Δt consists of exactly p guided bands. i) If fj = 0 on V01 , then, due to Proposition 2.2, {tζj } is a guided flat band of the Laplacian Δt on the perturbed graph Γ = Γ0 ∪ Γgt . Let fj = (fj,01 , fj,1 ) ∈ 2 (V1 ), where 0 = fj,01 ∈ 2 (V01 ) and fj,1 ∈ 2 (V1 \ V01 ). Using (3.7), we rewrite the fiber Laplacian Δt (ϑ), ϑ ∈ Td , for the Laplacian Δt acting on the perturbed graph Γ = Γ0 ∪ Γgt in the form Δt (ϑ) = P Δ0 (ϑ)P + tP1 Δ1 P1 = tKt (ϑ),

Kt (ϑ) = P1 Δ1 P1 + εP Δ0 (ϑ)P,

ε=

1 . t

We denote the eigenvalues of the operator Kt (ϑ) above its essential spectrum by Ep (ϑ, t)  . . .  E2 (ϑ, t)  E1 (ϑ, t),

ϑ ∈ Td .

(4.9)

Then the eigenvalue Ej (ϑ, t) of the operator Kt (ϑ) has the following asymptotics: Ej (ϑ, t) = ζj + ε Wj (ϑ) + O(ε2 )

(4.10)

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(see pp. 7–8 in [36]) uniformly in ϑ ∈ Td as t → ∞, where Wj (ϑ) = (0, fj,01 ), Δ0 (ϑ)(0, fj,01 ) V0c ,

(4.11)

· , · V0c denotes the inner product in 2 (V0c ). This yields the asymptotics of the eigenvalue λj (ϑ, t) of the operator Δt (ϑ): λj (ϑ, t) = t Ej (ϑ, t) = tζj + Wj (ϑ) + O(1/t).

(4.12)

+ Using this asymptotics for λ− j (t) = min λj (ϑ, t) and λj (t) = max λj (ϑ, t), we obtain ϑ∈Td

ϑ∈Td

± λ± j (t) = tζj + Wj + O(1/t), + where Wj± are defined in (2.20). Since sj (Δt ) = [λ− j (t), λj (t)], the asymptotics (4.12) also gives the second formula in (2.19). Using the formula (3.9) for the fiber Laplacian Δ0 (ϑ), we obtain

Wj (ϑ) = =

 

  Δ0 (ϑ)(0, fj,01 ) (v) f j (v) =

v∈V01



κv0 |fj (v)|2 −





 fj (v) − eiτ (e), ϑ fj (u) f j (v)

v∈V01 e=(v, u)∈Ac0



eiτ (e), ϑ fj (u) f j (v) (4.13)

e=(v, u)∈Ac0 v,u∈V01

v∈V01

=



κv0 |fj (v)|2 −



cosτ (e), ϑ fj (u) fj (v),

e=(v, u)∈Ac0 v,u∈V01

v∈V01

where κv0 is the degree of the vertex v on the unperturbed cylinder C0 and τ (e) ∈ Zd is the edge index defined by (3.3), (3.4). Using this formula we rewrite the constant Wj• defined in (2.20) in the form Wj• = max Ωj (ϑ) − min Ωj (ϑ), ϑ∈Td

ϑ∈Td



Ωj (ϑ) =

cosτ (e), ϑ fj (u) fj (v).

(4.14)

e=(v, u)∈Ac0 v,u∈V01 ,τ (e) =0

We have 

|Ωj (ϑ)| 

| cosτ (e), ϑ |  β01 ,

e=(v, u)∈Ac0 v,u∈V01 ,τ (e) =0

which yields Wj•  2β01 . ii) Let V01 = {v}. Then for Ωj (ϑ) defined in (4.14) we have Ωj (ϑ) = fj2 (v)



cosτ (e), ϑ ,

v)∈Ac0

e=(v, τ (e) =0

Using the identity



max Ωj (ϑ) = fj2 (v)β01 .

ϑ∈Td

(4.15)

cosτ (e), ϑ dϑ = 0 for each τ (e) = 0, we obtain

Td

−fj2 (v)β01  min Ωj (ϑ)  0. ϑ∈Td

(4.16)

Then (4.14)–(4.16) yield fj2 (v)β01  Wj•  2fj2 (v)β01 . From (4.15) and the condition fj (v) = 0 it follows that Ωj (·) = const iff β01 = 0. This yields the last statement of the item.

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Fig. 3. A finite graph Γ1 with p connected components Γ11 , . . . , Γ1p .

iii) If all positive eigenvalues ζp < . . . < ζ1 of the Laplacian Δ1 are distinct, then for t large enough all positive eigenvalues tζp < . . . < tζ2 < tζ1 of the Laplacian tΔ1 on the graph Γt satisfy tζp >  and t(ζj − ζj+1 ) >  for all j ∈ Np−1 . Then, due to Corollary 4.1, all guided bands of Δt are separated by gaps. Summing the second asymptotics in (2.19) over j = 1, . . . , p we obtain (2.22). iv) If there is no bridge connecting the vertices from the set V01 on the cylinder C, then for each j ∈ Np the function Wj defined by (4.13) is constant, i.e., Wj• = 0, and the second asymptotics in (2.19) and the asymptotics (2.22) take the form |sj (Δt )| = O(1/t) and |s(Δt )| = O(1/t), respectively. 2 Remark. The set Ac0 of all oriented edges of the cylinder C0 is infinite, but the sum in (4.13) is taken over a finite (maybe empty) set of edges (v, u) ∈ Ac0 for which v, u ∈ V01 . 4.3. Geometric properties of the guided spectrum Now we prove Proposition 2.2 and Corollary 2.5 about geometric properties of the guided spectrum for specific graphs. Proof of Proposition 2.2. Let ζ be an eigenvalue of the Laplacian Δ1 on Γ1 with an eigenfunction f ∈ 2 (V1 ) equal to zero on V01 . Then, due to (3.7), for each ϑ ∈ Td the function (0, f ) ∈ 2 (V c ) is an eigenfunction of the fiber Laplacian Δ(ϑ) associated with the same eigenvalue ζ. Thus, {ζ} is a guided flat band of the Laplacian Δ. 2 Remarks. 1) The ν1 × ν1 matrix Δ1 = {Δ1uv }u,v∈V1 associated to the Laplacian Δ1 on a finite graph Γ1 = (V1 , E1 ) in the standard orthonormal basis is given by Δ1uv = δuv κv1 − κuv ,

ν1 = #V1 ,

(4.17)

where δuv is the Kronecker delta, κv1 is the degree of the vertex v ∈ V1 on the graph Γ1 , κuv is the number of edges (u, v) ∈ A1 . 2) The sufficient conditions from Proposition 2.2 are equivalent to ζ being an eigenvalue of two operators: the Laplacian Δ1 and the Laplacian ΔD on Γ1 with Dirichlet boundary conditions f (v) = 0,

∀v ∈ V01 ,

(4.18)

i.e., ζ is an eigenvalue of two matrices: Δ1 = {Δ1uv }u,v∈V1 and its submatrix ΔD = {Δ1uv }u,v∈V1 \V01 . Proof of Corollary 2.5. i) Due to the connectivity of the periodic graph Γ0 , there exists a bridge on the cylinder C0 . First we consider the case when there exists a bridge-loop at some vertex u1 ∈ V0c . Due to the periodicity of the cylinder C0 , for each j ∈ Z we have uj = u1 + (j − 1)ad ∈ V0c , where ad is one of the periods of C0 . Let Γ1 = (V1 , E1 ) be a finite graph consisting of p ∈ N connected components Γ11 , . . . , Γ1p , where Γ1j is a graph consisting of two vertices uj ∈ V0c , vj ∈ / V0c and j edges connecting these vertices (see Fig. 3). The Laplacian Δ1 on the finite graph Γ1 has exactly p simple positive eigenvalues ζ1 = 2p,

ζ2 = 2(p − 1),

... ,

ζp = 2.

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The normalized eigenfunction fj ∈ 2 (V1 ) of the Laplacian Δ1 corresponding to the eigenvalue ζj has the form ⎧ 1 ⎪ ⎪ √2 , if v = uj ⎨ fj (v) = − √12 , if v = vj , j = 1, . . . , p. (4.19) ⎪ ⎪ ⎩ 0, otherwise Let Γt , t ∈ N, be a finite graph obtained from the graph Γ1 considering each edge of Γ1 to have the multiplicity t. Let Δt be the Laplacian on the perturbed graph Γ = Γ0 ∪ Γgt . Due to Theorem 2.4, for t large enough the guided spectrum of the Laplacian Δt consists of exactly p guided bands soj (Δt ) = sj (Δt ) separated by gaps and these bands satisfy |sj (Δt )| = Wj• + O(1/t),

j ∈ Np ,

(4.20)

where Wj• is defined in (2.20). Substituting the identities (4.19) into (4.13), we obtain the following expression for the function Wj : Wj (ϑ) =

κu0 j 2



−

cosτ (e), ϑ .

(4.21)

e=(uj , uj )∈Ac0

Due to the periodicity of the cylinder C0 , for each ϑ ∈ Td we have 



cosτ (e), ϑ =

e=(u1 , u1 )∈Ac0

cosτ (e), ϑ = . . . .

(4.22)

e=(u2 , u2 )∈Ac0

This and (4.21) yield that W1• = . . . = Wp• = 0. Then, due to (4.20), all guided bands are non-degenerate and |s(Δt )| =

p    sj (Δt ) = p W1• + O(1/t).

(4.23)

j=1

C , we obtain |s(Δt )| > C for t large enough. W1• Now let there be no bridge-loop on the cylinder C0 . Then the existing bridge connects some distinct vertices u1 , v1 ∈ V0c . Repeating the above arguments but with uj = u1 + (j − 1)ad ∈ V0c and vj = v1 + (j − 1)ad ∈ V0c , j ∈ Np , we also obtain the required statement. Note that in this case the function Wj has the form Choosing p >

Wj (ϑ) =

 1 0 κuj + κv0j + 2



cosτ (e), ϑ ,

j ∈ Np .

(4.24)

e=(uj , vj )∈Ac0

ii) Let there exist a vertex v ∈ V0c such that there is no bridge on C0 starting at v. We consider a finite graph Γt = (V1 , Et ) consisting of two vertices u ∈ / V0c and v and the edge (u, v) of multiplicity t ∈ N. The Laplacian on the finite graph Γt has one simple positive eigenvalue ζ1 = 2t. Let Δt be the Laplacian on the perturbed graph Γ = Γ0 ∪ Γgt . Due to Theorem 2.4, for t large enough the guided spectrum of Δt consists of exactly one guided band s1 (Δt ) and the length of this guided band satisfies |s1 (Δt )| = O(1/t). Thus, for any small ε > 0 there exists t ∈ N such that |s(Δt )| < ε. iii) This follows from (2.16) and the fact that the eigenvalues of the Laplacian on the finite graph Γt can be arbitrary large as t → ∞.

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iv) The sufficient conditions for the existence of guided flat bands are proved in Proposition 2.2. For examples of finite graphs Γ1 satisfying these conditions see Propositions 5.2, 5.3. 2 4.4. Reduction to operators on unperturbed cylinder We reduce the eigenvalue problem for the fiber Laplacian on the perturbed cylinder C to that on the unperturbed cylinder C0 . In order to do this we use the following well-known theorem [3]. Theorem 4.2 (Feshbach projection method). Let P be an orthogonal projection on a separable Hilbert space H, and let P ⊥ = 1 − P be its complement. Let T be a bounded self-adjoint operator. Assume that P ⊥ T P ⊥ is invertible on P ⊥ H. Then i) T is invertible on H if and only if its Feshbach map  −1 ⊥ F = P T P − P T P ⊥ P ⊥T P ⊥ P TP

(4.25)

is invertible on P H; in this case F −1 = P T −1 P ; ii) if T ψ = 0 for some vector 0 = ψ ∈ H, then FP ψ = 0, where P ψ = 0; iii) if Fϕ = 0 for some vector 0 = ϕ = P ϕ, then T ψ = 0, where   0 = ψ = P − P ⊥ (P ⊥ T P ⊥ )−1 P ⊥ T P ϕ; iv) the kernels of T and F have equal dimensions. Remark. The operator T in our consideration is Δ(ϑ) −λ, where Δ(ϑ) is the fiber Laplacian on the perturbed cylinder C = (V c , E c ). Proposition 4.3. Let P be the orthogonal projection of 2 (V c ) onto the subspace 2 (V0c ). Then the following statements hold true. i) If P ⊥ (Δ1 − λ)P ⊥ is invertible on P ⊥ 2 (V c ), then for each ϑ ∈ Td   λ ∈ σp Δ(ϑ)

⇔

  0 ∈ σp F(ϑ, λ) ,

(4.26)

and the kernels of Δ(ϑ) − λ and F(ϑ, λ) have equal dimensions. Here F(ϑ, λ) is the Feshbach map (4.25) for the operator Δ(ϑ) − λ, defined by (2.7), and F(ϑ, λ) has the form     −1 ⊥  F(ϑ, λ) = P Δ0 (ϑ) − λ P + P01 Δ1 − Δ1 P ⊥ P ⊥ (Δ1 − λ)P ⊥ P Δ1 P01 ,

(4.27)

P01 is the orthogonal projection of 2 (V c ) onto 2 (V01 ). ii) Let, in addition, a finite graph Γ1 consist of c := cΓ1 connected components Γ11 = (V11 , E11 ),

... ,

Γ1c = (V1c , E1c )

each of which has exactly one common vertex v1 , . . . , vc , respectively, with the unperturbed cylinder C0 = (V0c , E0c ). Then for each ϑ ∈ Td the Feshbach map (4.27) for the operator Δ(ϑ) − λ is just a fiber Schrödinger operator given by   F(ϑ, λ) = P Δ0 (ϑ) − λ P + Q(λ), where Q(λ) = Q(λ, ·) is a potential with the finite support V01 = {v1 , . . . , vc }:

(4.28)

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Fig. 4. a) The square lattice L2 ; the vertices from the fundamental vertex set V0c are big black points; the strip S is shaded; b) the cylinder C0 = L2 /Z (the edges of the strip are identified).

 −1 ⊥     Q(λ, vj ) = Pj Δ1j − Δ1j Pj⊥ Pj⊥ (Δ1j − λ)Pj⊥ Pj Δ1j Pj 2 ({vj }) ,

j = 1, . . . , c,

(4.29)

Δ1j is the Laplacian on the finite graph Γ1j , Pj is the orthogonal projection of 2 (V1j ) onto the onedimensional subspace 2 ({vj }). In particular, if the operator Δ1j − λPj⊥ is invertible, then the expression (4.29) can be written in the form   Q−1 (λ, vj ) = Pj (Δ1j − λPj⊥ )−1 Pj 2 ({vj }) .

(4.30)

iii) If P ⊥ (Δ1 − λ)P ⊥ is not invertible on P ⊥ 2 (V c ), then {λ} is a guided flat band of the Laplacian Δ on the perturbed graph Γ = Γ0 ∪ Γg1 . Proof. i) Let P1 be the orthogonal projection of 2 (V c ) onto 2 (V1 ). Then, using (4.25), (3.7) and the identities P P1 = P1 P = P01 ,

P P ⊥ = P ⊥ P = 0,

P ⊥ P 1 = P1 P ⊥ = P ⊥ ,

(4.31)

we have    −1 ⊥ F(ϑ, λ) = P Δ(ϑ) − λ P − P Δ(ϑ)P ⊥ P ⊥ (Δ(ϑ) − λ)P ⊥ P Δ(ϑ)P   = P P Δ0 (ϑ)P + P1 Δ1 P1 − λ P   −1 ⊥    P P Δ0 (ϑ)P + P1 Δ1 P1 P − P P Δ0 (ϑ)P + P1 Δ1 P1 P ⊥ P ⊥ (P Δ0 (ϑ)P + P1 Δ1 P1 − λ)P ⊥   −1 ⊥  P Δ1 P01 . = P Δ0 (ϑ) − λ P + P01 Δ1 P01 − P01 Δ1 P ⊥ P ⊥ (Δ1 − λ)P ⊥

(4.32)

This and Theorem 4.2 yield the required statement. ii) In this case Δ1 = Δ11 ⊕. . .⊕Δ1c . Then (4.27) has the form (4.28), (4.29). Let the operator Δ1j −λPj⊥ be invertible. Then, using the partitioned presentation of the inverse matrix (see p. 18 in [16]), we obtain (4.30). iii) Let P ⊥ (Δ1 − λ)P ⊥ not be invertible on P ⊥ 2 (V c ). Then λ is an eigenvalue of the operator P ⊥ Δ1 P ⊥ and, due to Proposition 2.2, {λ} is a guided flat band of the Laplacian Δ on the perturbed graph Γ. 2 5. Square lattice with guides In this section we consider the square lattice with specific guides. We obtain some properties of the guided spectrum on such graphs and give examples of the guided spectrum.

Let L2 = (V, E) be the square lattice, where the vertex set V = Z2 and the edge set E = (m, m +  (1, 0)), (m, m + (0, 1)), ∀ m ∈ Z2 , see Fig. 4.a. The Laplacian Δ0 on L2 has the form (Δ0 f )(m) = 4f (m) −

 |m−k|=1

f (k),

f ∈ 2 (Z2 ),

m ∈ Z2 .

(5.1)

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Fig. 5. a) The perturbed square lattice Γ = L2 ∪ Γg1 ; b) a finite graph Γ1 ; c) the perturbed cylinder C = Γ/Z.

We consider the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γg1 with a guide Γg1 . Due to Proposition 2.1, the Laplacian Δ on Γ has the decomposition (2.6), (3.7) into a constant fiber direct integral, where the fiber unperturbed Laplacian Δ0 (ϑ) acts on f ∈ 2 (Z) and is given by Δ0 (ϑ) = 2(1 − cos ϑ) + h, (hf )(n) = 2f (n) − f (n + 1) − f (n − 1),

n ∈ Z,

(5.2)

for all ϑ ∈ T = (−π, π]. It is well known that the spectrum of the Laplacian h on Z is given by σ(h) = σac (h) = [0, 4]. Then, due to Proposition 2.1.ii), the spectrum of each fiber Laplacian Δ(ϑ), ϑ ∈ T, has the form       σ Δ(ϑ) = σac Δ(ϑ) ∪ σp Δ(ϑ) ,     σac Δ(ϑ) = σac Δ0 (ϑ) = [2 − 2 cos ϑ, 6 − 2 cos ϑ],

(5.3)

  σp Δ(ϑ) is the set of all eigenvalues of Δ(ϑ) of finite multiplicity given by (2.10). The spectrum of the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γg1 has the form σ(Δ) = σ(Δ0 ) ∪ s(Δ),

σ(Δ0 ) = [0, 8],

s(Δ) =

N 

sj (Δ) = sac (Δ) ∪ sf b (Δ),

j=1

where N is defined in (2.11). Now we consider the perturbed square lattice Γ = L2 ∪Γg1 when a finite graph Γ1 has exactly one common vertex v1 = 0 with the unperturbed cylinder C0 = L2 /Z = (Z, E0c ), see Fig. 5. Proposition 5.1. Let Γ1 = (V1 , E1 ) be a finite connected graph and let V01 = Z ∩ V1 = {0}. Then the absolutely continuous guided spectrum sac (Δ) and the flat band guided spectrum sf b (Δ) of the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γg1 satisfy    sac (Δ) = λ ∈ R : 2  λ − 4 + Q2 (λ)  6, Q(λ) > 0 ,

 sf b (Δ) = λ ∈ R : λ is an eigenvalue of the operator P ⊥ Δ1 P ⊥ ,

(5.4) (5.5)

where P is the orthogonal projection of 2 (V1 ) onto the one-dimensional subspace 2 ({0}), Δ1 is the Laplacian on the finite graph Γ1 ,    −1 ⊥   Q(λ) = P Δ1 − Δ1 P ⊥ P ⊥ (Δ1 − λ)P ⊥ P Δ1 P 2 ({0}) .

(5.6)

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In particular, if the operator Δ1 − λP ⊥ is invertible, then the expression (5.6) can be written in the form   Q−1 (λ) = P(Δ1 − λP ⊥ )−1 P 2 ({0}) .

(5.7)

Proof. Let λ ∈ R not be an eigenvalue of the operator P ⊥ Δ1 P ⊥ . For each ϑ ∈ T, using (4.28), (4.29) and (5.2), we obtain the Feshbach map for the operator Δ(ϑ) − λ:   F(ϑ, λ) = P Δ0 (ϑ) − λ P + Q(λ) = P (h − μ)P + Q(λ), (5.8) μ = λ − (2 − 2 cos ϑ), where P is the orthogonal projection of 2 (V c ) onto the subspace 2 (Z), h is the Laplacian on Z, the potential Q(λ) has the support {0} and is given by (5.6) or, in the case when Δ1 − λP ⊥ is invertible, by (5.7). Then, due to Proposition 4.3 and the identity (5.8), we have   λ ∈ σp Δ(ϑ)

⇔

  0 ∈ σp F(ϑ, λ)

⇔

  μ ∈ σp h + Q(λ) .

(5.9)

It is well known that the Schrödinger operator h + Q on Z with a potential Q having a support at a single point has the unique eigenvalue  μ=

2+ 2−

 

4 + Q2 ,

Q > 0,

4 + Q2 ,

Q < 0.

(5.10)

Combining this with (5.9) and using the second identity in (5.8) and the fact that any eigenvalue λ of Δ(ϑ) satisfies λ  2 − 2 cos ϑ, we obtain    σp Δ(ϑ) = λ ∈ R : λ − 4 + Q2 (λ) = 4 − 2 cos ϑ,

 Q(λ) > 0 ,

(5.11)

which, due to the definition (2.12) of the guided spectrum, yields (5.4) and λ ∈ / sf b (Δ). Let λ ∈ R be an eigenvalue of the operator P ⊥ Δ1 P ⊥ . Then, due to Proposition 2.2, {λ} is a guided flat band of the Laplacian Δ on Γ. Thus, sf b (Δ) has the form (5.5). 2 Remark. If Q(λ) < 0 for some λ > 8, then λ lies in the gap of the spectrum of the Laplacian Δ on the perturbed square lattice. 5.1. Lattice with p pendant edges at each vertex of Z × {0} Let Γ1 = (V1 , E1 ) be a star graph of order p + 1, i.e., a tree on p + 1 vertices v1 , v2 , . . . , vp+1 with one vertex v1 = 0 having degree p and the other p having degree 1. We assume that V01 = Z ∩ V1 = {0}

(5.12)

and consider the perturbed square lattice Γ = L2 ∪ Γg1 , see Fig. 6. Proposition 5.2. The guided spectrum of the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γg1 , where Γ1 is a star graph of order p + 1 satisfying (5.12), has the form s(Δ) = sac (Δ) ∪ sf b (Δ),

+ sac (Δ) = s1 (Δ) = [λ− 1 , λ1 ],

sf b (Δ) = s2 (Δ) ∪ . . . ∪ sp (Δ) = {1},

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Fig. 6. a) The perturbed square lattice Γ = L2 ∪ Γg1 ; b) the perturbed cylinder C = Γ/Z.

where the guided flat band {1} has the multiplicity p − 1 and s1 (Δ) ⊂ [p + 3, p + 8], λ− 1

= p + 3 + O(1/p),

λ+ 1

|s1 (Δ)| < 4,

= p + 7 + O(1/p),

as

p → ∞.

(5.13)

Moreover, lim |s1 (Δ)| = 2β+ = 4, where β+ is defined in (2.18). p→∞

Remark. For example, for p = 1 and p = 2 the guided band s1 (Δ) has the form s1 (Δ) ≈ [4,38; 8,30] and s1 (Δ) ≈ [5,18; 9,01], respectively. Proof. The (p + 1) × (p + 1) matrix Δ1 = {Δ1uv }u,v∈V1 defined by (4.17) for the Laplacian Δ1 on the star-graph Γ1 has the form   p b Δ1 = (5.14) , b = (−1, . . . , −1) ∈ Cp , b∗ Ip where Ip is the identity p × p matrix. Since λ = 1 is an eigenvalue of the matrix Δ1 of multiplicity p − 1, due to Proposition 2.2, the Laplacian Δ on Γ has the guided flat band {1} of multiplicity p − 1. Now we find the absolutely continuous guided spectrum sac (Δ). Since the number of guided bands N  p and the guided flat band {1} has multiplicity p−1, the absolutely continuous guided spectrum sac (Δ) consists pλ + of at most one band s1 (Δ) = [λ− (5.4), where Q(λ) = λ−1 . 1 , λ1 ] and, due to Proposition 5.1, has the form 2 The inequality Q(λ) > 0 gives that λ > 1. The graphs of the function f (λ) = 4 + Q (λ) for λ > 1 and of the functions g1 (λ) = λ − 2, g2 (λ) = λ − 6 are shown in Fig. 7. Thus, the absolutely continuous guided + + − spectrum sac (Δ) consists of exactly one guided band s1 (Δ) = [λ− 1 , λ1 ] and |s1 (Δ)| = λ1 −λ1 < 4, see Fig. 7. − − − From the identity f (λ1 ) = g1 (λ1 ) we obtain that λ1 is a root of the equation h(λ) ≡ λ3 − 6λ2 + (9 − p2 )λ − 4 = 0.

(5.15)

Using that h(p + 3) = −4 < 0,

h(p + 4) = 2p2 + 9p > 0,

we obtain that the equation (5.15) has a unique solution λ− 1 ∈ [p + 3, p + 4]. This and |s1 (Δ)| < 4 give that s1 (Δ) ⊂ [p + 3, p + 8].

(5.16)

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Fig. 7. The graphs of the functions f , g1 , g2 . 2 We introduce the new variable α = λ− 1 − p, α ∈ [3, 4]. Then, after dividing by p , the equation (5.15) has the form

2α − 6 + O(1/p) = 0,

as

p → ∞,

which yields α = 3 + O(1/p) and λ− 1 = p + 3 + O(1/p) as p → ∞. Thus, the first asymptotics in (5.13) has been proven. Similarly, we obtain that λ+ 1 = p + 7 + O(1/p) as p → ∞. These asymptotics yield that |s1 (Δ)| → 4 as p → ∞. On the other hand, the number β+ = 2, since for each vertex of the cylinder C there are two bridges starting at this vertex. Thus, we have lim |s1 (Δ)| = 2β+ = 4. 2 p→∞

5.2. Lattice with mandarin guides An s-mandarin graph is a graph consisting of two vertices and s edges connecting these vertices. Let Γ1 = (V1 , E1 ) be a union of two s-mandarin graphs with one common vertex v1 = 0, see Fig. 2.b. We assume that V01 = Z ∩ V1 = {0}

(5.17)

and consider the perturbed square lattice Γ = L2 ∪ Γg1 , see Fig. 2. Proposition 5.3. The guided spectrum of the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γg1 , where Γ1 is a union of two s-mandarin graphs with one common vertex satisfying (5.17), has the form s(Δ) = sac (Δ) ∪ sf b (Δ),

+ sac (Δ) = s1 (Δ) = [λ− 1 , λ1 ],

sf b (Δ) = s2 (Δ) = {s},

where s1 (Δ) ⊂ [3s + 1, 3s + 6] for s  2, 4 + O(1/s), λ+ λ− 1 = 3s + 1 = 3s + 4 + O(1/s), 3

as

s → ∞.

Remarks. 1) For example, for s = 1 and s = 2 the guided band s1 (Δ) has the form s1 (Δ) ≈ [5,18; 9,01] and s1 (Δ) ≈ [7,75; 11,26], respectively.

(5.18)

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Fig. 8. The graphs of the functions f , g1 , g2 .

2) For s  8 the guided flat band {s} is embedded into the absolutely continuous spectrum of the Laplacian Δ. For s > 8 it lies in a gap. Proof. The matrix Δ1 = {Δ1uv }u,v∈V1 defined by (4.17) for the Laplacian Δ1 on the graph Γ1 has the form  Δ1 = s

 2 b∗

b I2

,

b = (−1, −1).

(5.19)

Since λ = s is a simple eigenvalue of the matrix Δ1 , due to Proposition 2.2, the Laplacian Δ on Γ has a guided flat band {s}. + The absolutely continuous guided spectrum sac (Δ) consists of at most one band s1 (Δ) = [λ− 1 , λ1 ] and, 2sλ due to Proposition 5.1, has the form (5.4), where Q(λ) = λ−s . The inequality Q(λ) > 0 gives that λ > s.  The graphs of the function f (λ) = 4 + Q2 (λ) for λ > s and of the functions g1 (λ) = λ − 2, g2 (λ) = λ − 6 are shown in Fig. 8. Thus, the absolutely continuous guided spectrum sac (Δ) consists of exactly one guided − + + band s1 (Δ) = [λ− 1 , λ1 ] and |s1 (Δ)| = λ1 − λ1 < 4, see Fig. 8. − − From the identity f (λ1 ) = g1 (λ1 ) we obtain that λ− 1 is a root of the equation h(λ) ≡ (λ − 4)(λ − s)2 − 4s2 λ = 0.

(5.20)

Using that h(3s + 1) = −4s2 − 9s − 3 < 0,

h(3s + 2) = 4(2s2 − s − 2) > 0

for s  2,

we obtain that for s  2 the equation (5.20) has a unique solution λ− 1 ∈ [3s + 1, 3s + 2].

(5.21)

This and |s1 (Δ)| < 4 give that s1 (Δ) ⊂ [3s + 1, 3s + 6]. 2 We introduce the new variable α = λ− 1 − 3s, α ∈ [1, 2]. Then, after dividing by s , the equation (5.20) has the form 12α − 16 + O(1/s) = 0,

as

s → ∞,

4 which yields α = 43 + O(1/s) and λ− 1 = 3s + 3 + O(1/s) as s → ∞. Thus, the first asymptotics in (5.18) + has been proven. Similarly, we obtain that λ1 = 3s + 4 + O(1/s) as s → ∞. 2

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Fig. 9. a) The perturbed square lattice Γ = L2 ∪ Γgt ; b) the perturbed cylinder C = Γ/Z.

5.3. Lattice with path guides Let Γ1 = (V1 , E1 ) be a path of length 2, i.e., a connected graph with two vertices v1 and v3 having degree 1 and a vertex v2 having degree 2 and let Γt = (V1 , Et ) be a finite graph obtained from the graph Γ1 considering each edge of Γ1 to have the multiplicity t. We assume that V01 = Z ∩ V1 = {v1 = 0} and consider the perturbed square lattice Γ = L2 ∪ Γgt , see Fig. 9. Proposition 5.4. The guided spectrum of the Laplacian Δ on the perturbed square lattice Γ = L2 ∪ Γgt has the form  s(Δ) = sac (Δ) =

s1 (Δ),

if t = 1, 2

s1 (Δ) ∪ s2 (Δ),

if t  3

,

+ where ss (Δ) = [λ− s , λs ], s = 1, 2,

λ− 1 = 3t +

1 3

λ+ 1 = 3t + 1 + O(1/t),

+ O(1/t),

λ− 2 = t + 1 + O(1/t),

as

λ+ 2 = t + 3 + O(1/t),

t → ∞.

(5.22)

Proof. The matrix Δ1 = {Δ1uv }u,v∈V1 defined by (4.17) for the Laplacian Δ1 on the graph Γt has the form  Δ1 = t



 1 b∗

b ΔD

,

b = (−1, 0),

ΔD =

2 −1 −1 1

 .

(5.23)

Since the matrices Δ1 and ΔD have no equal eigenvalues, due to Proposition 5.1, the Laplacian Δ on Γ has no guided flat bands. Then the absolutely continuous guided spectrum sac (Δ) consists of at most two bands and, due to Proposition 5.1, has the form (5.4), where Q(λ) =

tλ(λ − 2t) . λ2 − 3tλ + t2

The inequality Q(λ) > 0 gives that λ∈

 3−√5 2

  √  t, 2t ∪ 3+2 5 t, +∞ .

(5.24)

√  For t  4 the graphs of the function f (λ) = 4 + Q2 (λ) for λ > 3−2 5 t and of the functions g1 (λ) = λ − 2, g2 (λ) = λ − 6 are shown in Fig. 10. Thus, for t  4 the absolutely continuous guided spectrum sac (Δ) + consists of exactly two guided bands ss (Δ) = [λ− s , λs ], s = 1, 2. Since the Laplacian Δ1 on the graph Γt has the positive eigenvalues ζ1 = 3t and ζ2 = t, Corollary 4.1 gives that s1 (Δ) ⊂ [3t, 3t + 8], s2 (Δ) ⊂ [t, t + 8].

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Fig. 10. The graphs of the functions f , g1 , g2 ; t  4. − − We introduce the new variable α = λ− 1 − 3t, α ∈ [0, 8]. Then from the equation f (λ1 ) = g1 (λ1 ) it follows that

36α − 12 + O(1/t) = 0,

as

t → ∞,

1 which yields α = 13 + O(1/t) and λ− 1 = 3t + 3 + O(1/t) as t → ∞. Similarly, we obtain the remaining asymptotics in (5.22). From (5.4) by a direct calculation we obtain the absolutely continuous guided spectrum of the Laplacian Δ for t  3:

⎧ ⎪ s1 (Δ) ≈ [4,47; 8,31], ⎪ ⎪ ⎨ sac (Δ) = s1 (Δ) ≈ [6,66; 9,45], ⎪ ⎪ ⎪ ⎩ s2 (Δ) ∪ s1 (Δ) ≈ [4,62; 6] ∪ [9,51; 11,44],

t=1 t=2 . t=3

2

Acknowledgment Our study was supported by the RSF grant No. 15-11-30007. References [1] K. Ando, Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice, Ann. Henri Poincaré 14 (2013) 347–383. [2] K. Ando, H. Isozaki, H. Morioka, Spectral properties of Schrödinger operators on perturbed lattices, Ann. Henri Poincaré 17 (2016) 2103–2171. [3] V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (2) (1998) 299–395. [4] G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, AMS, 2013. [5] A. Boutet de Monvel, J. Sahbani, On the spectral properties of discrete Schrödinger operators: the multi-dimensional case, Rev. Math. Phys. 11 (1999) 1061–1078. [6] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A. Geim, The electronic properties of graphene, Rev. Modern Phys. 81 (2009) 109–162. [7] D.J. Colquitt, M.J. Nieves, I.S. Jones, A.B. Movchan, N.V. Movchan, Localization for a line defect in an infinite square lattice, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013) 20120579. [8] M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298–305. [9] D. Gieseker, H. Knörrer, E. Trubowitz, The Geometry of Algebraic Fermi Curves, Perspectives in Mathematics, vol. 14, Academic Press, Inc., Boston, MA, 1993. [10] B. Güneysu, Semiclassical limits of quantum partition functions on infinite graphs, J. Math. Phys. 56 (2) (2015) 022102. [11] P. Harris, Carbon Nano-Tubes and Related Structure, Cambridge University Press, Cambridge, 2002.

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